Abstract: Born Infeld inspired gravity has been proposed recently in cosmological context. The nice feature of this theory is that it reduces exactly to standard General Relativity when no matter is present. However, inside matter distributions, the theory differs from Einstein's theory. In that sense, it is more a modification to the matter / geometry coupling rather than a modification of gravity itself.
We start reviewing the theory and present how the Post-Newtonian expansion is affected by the modification to General Relativity. Then, we argue that in the Newtonian regime there is no singularity forming during the collapse of a spherically symmetric dust distribution. Rather, we give strong evidences that the end-point of the gravitational collapse is a regular, "pressureless" star. We discuss the latter and other compact star solutions, both in the non-relativistic and in the relativistic theory. We conclude by discussing open issues and future work.